De Ratiociniis in Ludo Aleae [in:] Exercitationum Mathematicarum libri quinque...

Leiden: Elzevir, 1657.

First edition of Huygens' De ratiociniis in aleae ludo, the first printed treatise on probability. “Having heard in Paris about Pascal’s work in probability problems, Huygens (1629-95) himself took up their study in 1656. This resulted in the Tractatus de ratiociniis …, a treatise that remained the only book on the subject until the eighteenth century. In his first theorems Huygens deduced that the ‘value of a chance,’ in the case where the probabilities for a and b are to each other as p:q, is equal to (pa + qb) / (p + q). He thus introduced as a fundamental concept: the expectation of a stochastic variable rather than the probability of a process (to put it in modem terms). Subsequent theorems concern the fair distribution of the stakes when a game is broken off prematurely. The treatise closes with five problems, the last of which concerns expected duration of play” (DSB). “If one looks on the first calculations of a probability by Cardan and by Galileo as unimportant and says that the real begetter of the calculus of probabilities is he who first put it on a sound footing, then we should pass over not only Cardan and Galileo but also Pascal and Fermat. For although Fermat, if provoked, could have done as much and more than his successors, yet the fact remains that his contribution was in effect the extension of the idea of the exhaustive enumeration of the fundamental probability set, which had already been given by Galileo. The scientist who first put forward in a systematic way the new propositions evoked by the problems set to Pascal and Fermat, who gave the rules and who first made definitive the idea of mathematical expectation was Christianus Huygens” (David, p. 110). “It was not until the tremendous researches of the 1690-1710 period which resulted in Essai d'Analyse sur les Jeux de Hasard (Montmort, 1708), Ars Conjectandi (James Bernoulli, 17l3), Calcul des Chances, à la statistique générale, àla statistique des décès et aux rentes viagères (Nicholas Struyck, 1713), and Doctrine of Chances (Abraham de Moivre, 1718), that Huygens' work was superseded, and then not entirely since James Bernoulli incorporated it in the Ars Conjectandi” (ibid., pp. 115-116). Huygens’ treatise was appended to Schooten’s Exercitationum, which comprises five treatises, each with their own title-page, on various mathematical subjects. This was Schooten’s principal original work. “Book I contains elementary arithmetic and geometry problems … Book II is devoted to constructions using straight lines only and Book III to the reconstruction of Apollonius’ Plane Loci on the basis of hints given by Pappus. Book IV is a revised version of Schooten’s treatment of the kinematic generation of conic sections, and Book V is a collection of individual problems. Worth noting, in particular, is the restatement of Hudde’s method for the step-by-step building-up of equations for angular sections and the determination of the girth of the folium of Descartes: x3 + y3 = 3axy. Also noteworthy is the determination of Heronian triangles of equal perimeter and equal area (Roberval’s problem) according to Descartes’s method (1633)” (ibid.).

Around 1564, Girolamo Cardano composed a work on probability entitled Liber de ludo aleae (‘Book on Games of Chance’), but this was not published until 1663 as part of his Opera. Some time between 1613 and 1623, Galileo wrote about die-throwing problems but this remained unpublished until the 20th century. Italian writers of the fifteenth and sixteenth centuries, notably Pacioli, Tartaglia, and Cardano, had discussed the problem of the division of a stake between two players whose game was interrupted before its close (the so-called ‘Problem of Points’). The problem was proposed to Pascal and Fermat, probably in 1654, by the gambler Chevalier de Méré. The ensuing correspondence between the two men is generally regarded as fundamental to the development of modern concepts of probability, but it was not published until 1679, in Fermat’s Varia opera mathematica (some of Pascal’s work on probability also appeared in his Traité du triangle arithmetique of 1665).

“Huygens visited Paris for the first time in the autumn of 1655. He met Roberval and Mylon, a friend of Carcavi, but did not meet Carcavi and Pascal. According to himself he heard about the probability problems discussed the year before but was not informed about the methods used and the solutions obtained. This sounds odd in view of the fact that Roberval was fully informed about the Pascal-Fermat correspondence; however, Roberval does not seem to have been much interested in probability theory. On his return to Holland Huygens solved the problems, and in April 1656 he sent van Schooten his manuscript of the treatise. At the same time he wrote to Roberval and asked for his solution of the most difficult problem (Proposition 14) in the treatise for comparison with his own. Van Schooten proposed that Huygens' treatise would be published at the end of his Exercitationum Mathematicarum, which he was preparing for publication. This would require a translation of the treatise into Latin, and after some discussion van Schooten offered to do this himself.

“In the meantime, since Roberval did not answer his letter, Huygens wrote to Mylon who, through Carcavi, sent the problem on to Fermat. In a letter from Fermat to Carcavi and from him to Huygens (22 June 1656), Fermat gave the solution without proof, a solution which to Huygens' satisfaction agreed with his own. Furthermore, Fermat posed five problems to Huygens, which Huygens immediately solved. He sent the solutions to Carcavi on 6 July and asked him to inform Mylon, Pascal, and Fermat to determine whether their solutions agreed with his. Huygens later used two of Fermat's problems as problems 1 and 3 at the end of his treatise. Carcavi’s answer of 28 September convinced Huygens that the methods employed by Pascal and himself were in agreement. Further, this letter contained a problem posed by Pascal to Fermat, which Huygens solved and included as the fifth problem in his treatise. Carcavi’s letter contained the solutions given by Fermat and Pascal without the proofs. Huygens' solution, also without proof, is stated in his letter to Carcavi on 1 2 October.

“In March 1657 van Schooten sent Huygens the Latin version for final additions and corrections. Huygens then added Proposition 9 and five problems for the reader, among them the three by Fermat and Pascal mentioned above. In a letter to van Schooten, which was used as Huygens' preface, Huygens stressed the importance of this new topic and stated that ‘for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention.’ The De Ratiociniis in Ludo Aleae was published in September 1657; the Dutch version was not published until 1660 …

“The De Ludo Aleae is composed as a modern paper on probability theory. From an axiom on the value of a fair game Huygens derives three theorems on expectations. He uses the theorems to solve a number of problems of current interest on games of chance by recursion. Finally, he poses five problems, gives the answers to three of them, and leaves the proofs to the reader. The three theorems and the eleven problems are usually called Huygens’ 14 propositions” (Hald, pp. 67-69).

“Huygens’ fourteen propositions run as follows:

I: To have equal chances of getting a and b is worth (a + b)/2.

II: To have equal chances of getting a, b or c is worth (a + b + c)/3.

III: To have p chances of obtaining a and q of obtaining b, chances being equal, is worth (pa + qb)/p + q.

IV: Suppose I play against an opponent as to who will win the first three games and that I have already won two and he one. I want to know what proportion of the stakes is due to me if we decide not to play the remaining games.

V: Suppose that I lack one point and my opponent three. What proportion of the stakes, etc.

VI: Suppose that I lack two points and my opponent three, etc.

VII: Suppose that I lack two points and my opponent four, etc.

VIII: Suppose now that three people play together and that the first and second lack one point each and the third two points.

IX: In order to calculate the proportion of stakes due to each of a given number of players who are each given numbers of points short, it is necessary, to begin with, to consider

what is owing to each in turn in the case where each might have won the succeeding game.

X: To find how many times one may wager to throw a six with one die.

XI: To find how many times one should wager to throw 2 sixes with 2 dice.

XII: To find the number of dice with which one may wager to throw 2 sixes at the first throw.

XIII: On the hypothesis that I play a throw of 2 dice against an opponent with the rule that if the sum is 7 points I will have won but that if the sum is 10 he will have won, and that we split the stakes in equal parts if there is any other sum, find the expectation of each of us.

XIV: If another player and I throw turn and turn about with 2 dice on condition that I will have won when I have thrown 7 points and he will have won when he has thrown 6, if I let him throw first find the ratio of my chance to his …

“Huygens’ treatise ends with the words ‘I finish by stating some propositions’, by which he means that he sets out some exercises and gives the answers in an appendix. James Bernoulli set out the solutions in the Ars Conjectandi. These exercises are more interesting than the formal propositions because they are possibly the first time that sampling with and without replacement occurs in the literature. Exercise I is just the problem of points all over again, but Exercise I1 runs as follows:

Three gamblers A, B and C take 12 balls of which 4 are white and 8 black. They play with the rules that the drawer is blindfold, A is to draw first, then B and then C, the winner to be the one who first draws a white ball. What is the ratio of their chances?

“Huygens, as can be seen from his answer, imagines that each ball is replaced after drawing. James Bernoulli pointed out that there were three possible interpretations of the exercise and solved all three. These were (i) the ball can be replaced after drawing, (ii) if it is not replaced after drawing then it may be supposed that the gamblers start with a central pool of 12 balls, or (iii) that they each have 12 balls. A mathematician called John Hudde wrote to Huygens in 1665 pointing out the second interpretation. Montmort, de Moivre and Struyck each discuss the different models. It is not known from where Huygens derived this exercise, and he may have made it up for himself. The third exercise is one of Fermat's problems:

A wagers B that, given 40 cards of which 10 are of one colour, 10 of another, 10 of another and 10 of another, he will draw 4 so as to have one of each colour.

“Exercise IV is again capable of two interpretations (Huygens samples with replacement), while Exercise V is again the problem of points. There is, however, in this exercise possibly the first hint of the gambler's ruin problem, since A and B start with 12 balls and continue to throw three dice on the condition that if 11 is thrown A gives a ball to B and if 14 is thrown B gives one to A. The game is to continue until one or other has all the balls” (David, pp. 116-119).

“Frans van Schooten (1615-60) established a vigorous research school in Leiden which included his private pupils Christiaan Huygens, Henrik van Heuraet and Johannes Hudde, and this school was one of the main reasons for the rapid development of Cartesian geometry in the mid 17th-century. It had been on Descartes’ recommendation that van Schooten replaced Jan Stampioen as tutor to Huygens and his brother. The teacher and his extremely talented pupil soon became good friends. After his students had left Leiden, van Schooten, Huygens, Hudde and van Heuraet corresponded regarding the properties of curves and other topics at the forefront of research at the time. Most of the correspondence was directed through van Schooten in the sense that his students would write to him to explain their discoveries and he would inform the others as well as publish certain of their results as appendices to his own publications. In a small way he was copying Mersenne’s way of operating which he had experienced at first hand in Paris” (MacTutor).

In 1649 Schooten published the first separate, and first Latin edition, of Descartes’ La Géométrie, which he greatly enlarged in the second edition of 1659-61. “The mathematical community learned about the wealth of Descartes’s new ideas through the works of van Schooten ... In the second edition the commentaries were enlarged, and van Schooten included the work by his students van Heuraet, Hudde, Huygens and de Witt. This edition served as the basic textbook for the generation that, in the last quarter of the century, took the lead in introducing differential and integral calculus” (Jahnke).

“This 1659-1661 edition contained appendices by three of van Schooten disciples, Jan de Witt, Johan Hudde, and Hendrick van Heuraet. He had earlier published De organica conicarum sectionum in plano descriptione, tractatus (1646) which was reprinted as part of his major work Exercitationes mathematicae libri quinque (Mathematical exercises in five books) in 1657 … The five books of Exercitationes mathematicae each have about 100 pages, so the whole work is quite a large treatise. Book I gives a review of arithmetic and basic geometry. In Book II, Constructio problematium simplicium geometricorum, he studies straight edge constructions for solving geometrical problems, followed by an attempted reconstruction of the works of Apollonius on plane loci in Book III. In Book IV, Organica conicarum sectionum, which contains the earlier publication, he studied the problem of drawing conic sections mechanically. He gave the, now standard, method of drawing an ellipse with a piece of string whose ends are attached to two pegs. Book V is titled Sectiones triginta miscellaneas and develops combinatorial techniques for counting problems. Fermat and Descartes had discovered new pairs of amicable numbers some years before (pairs of numbers, the sum of the factors of one number being equal to the other number). In this book van Schooten gives a methods to find these numbers which he hoped would lead to the discovery of further pairs of amicable numbers” (MacTutor).

Bierens de Haan 4222; Honeyman 2808; Sotheran, First Supplement 1767; Parkinson p. 94. David, Games, Gods and Gambling, 1962. Hald, A History of Probability and Statistics and their Applications before 1750, 2003. Jahnke, A History of Analysis, 2003.

4to (195 x 146 mm), pp. [xii], 534, [2], with numerous diagrams in text. Contemporary calf, raised bands and gilt spine, ex-libris to front paste down, a very fine and unrestored copy.

Item #5784

Price: $25,000.00